A system of equations has infinitely many solutions when the equations are dependent, meaning they represent the same line or plane (in higher dimensions). This essentially means they are multiples of each other and overlap completely.
Understanding Infinite Solutions
Infinite solutions occur when the equations in a system are not independent. Instead of intersecting at a single point (one solution) or being parallel and never intersecting (no solution), they are essentially the same equation presented in a different form. This results in the solution set being the entire line (or plane, etc.) represented by the equation.
Conditions for Infinite Solutions
The key condition for a system of equations to have infinitely many solutions is that one equation is a scalar multiple of the other (in a system of two equations). This means you can multiply one entire equation by a constant to get the other equation. Here's a breakdown:
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Two Linear Equations: In a system of two linear equations, infinite solutions exist when the two equations represent the same line. Graphically, this means the lines are coincident (they overlap). Algebraically, this means one equation can be transformed into the other by multiplying it by a constant. This implies they have the same slope and y-intercept.
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More than Two Equations: The concept extends to systems with more than two equations. For infinite solutions, the equations must be consistent and represent overlapping geometrical objects (lines, planes, hyperplanes). In this case, the rank of the augmented matrix of the system will be less than the number of variables.
Examples
Let's consider a simple system of two linear equations:
2x + y = 4
4x + 2y = 8Notice that the second equation is simply the first equation multiplied by 2. If you were to graph these equations, they would be the exact same line. Therefore, any (x, y) pair that satisfies the first equation will also satisfy the second, leading to an infinite number of solutions. For instance, (0, 4), (1, 2), and (2, 0) are all solutions to both equations.
Contrast with One Solution and No Solution
To further illustrate, let's compare to situations with one solution and no solution:
| Scenario | Description | Example |
|---|---|---|
| One Solution | Lines intersect at one point; equations are independent. | x + y = 3, x - y = 1 |
| No Solution | Lines are parallel and do not intersect; equations are inconsistent. | x + y = 3, x + y = 5 |
| Infinite Solutions | Lines are coincident (same line); equations are dependent. | x + y = 3, 2x + 2y = 6 |
Conclusion
A system of equations has infinitely many solutions when the equations are dependent, essentially representing the same relationship or geometrical object. This dependence allows any solution to one equation to also be a solution to the others, creating an infinite solution set.
